Though there are quite many questions on MSE about how to learn algebraic geometry, personally I still have something not very clear. Actually, I feel good when learning algebra and have studied abstract algebra (basic group, ring, module, field theory) and some commutative algebra from Atiyah's well-known book (mainly Ch1-9), but I think I'm lack of geometric intuition and meet some challenges when learning differential manifolds. So I prefer starting learning AG from a algebraic viewpoint or in an algebraic-oriented approach and supplying with the geometric meaning of these algebraic results (Of course I don't think AG is just somewhat "advanced commtutative algebra").
I have tried some books in AG. I find the last chapter of Patrick Morandi's Field and Galois Theory (GTM 167) is very helpful, which introduces some fundamental facts of AG, such as the definition of algebraic variety, the Nullstellensatz (without proof), the dimension of a variety and some related results. So does the first two chapters of Karen E. Smith's book An Invitation to Algebraic Geometry, but this book is a bit concise, not like GTM 167, which contains a lot of details. Besides, someone tells me that one can start learning AG from algebraic curves. So I tried Frances Kirwan's book Complex Algebraic Curves, finishing reading the first 4 chapters. This book is quite elementary, which assumes only complex analysis and some results from topology and no CA. I like its introduction of complex projective spaces, which is well motivated. However, since no CA is involved, the definition of intersection mutiplicity and the proof of Bezout theorem is too elementary (as far as I am concerned, some books define the intersection mutiplicity as the dimension of a certain ring, this seems like a modern definition), so I don't think I should remember all the details of this proof in the book and I think there may exist a more general result in modern AG. Also, I find William Fulton's Algebaic Curves is very popular and indeed uses a more algebraic approach. But I doubt that it maybe too concise since some proofs are left as exercises. I wonder if it's friendly enough for a self-learner like me. Other than all these books mentioned above, I find Klaus Hulek's Elementary Algebraic Geometry is attractive to me since it's similar to Fulton's book and contains a lot of details.
My questions are: How can I start learning algebraic geometry from an algebraic viewpoint? For now, should I complete all of exercises in Atiyah or finish Fulton's book and do all the exercises? Besides, is it possible to begin studying AG from an algebraic-oriented approach and supply with some geometric meaning? Someone tells me that one should know a lot about algebraic topology, differential geometry, Riemann surface before studying AG, but others say that a good understanding of CA and HA are enough to learn modern AG (at the begining point), such as GTM52. I hope someone can give me some suggestions. Many thanks.
P.S. My Maths background:
Analysis: Real Analysis from Royden (Part 1 and most of Part 2) and Big Rudin (Ch 1-8); Complex analysis from Conway's GTM11 (Ch1-6, before the Riemann mapping thm); Basic ODE, PDE; Only a bit knowledge of FA (elementary Banach spaces, Hilbert spaces).
Algebra: Abstract algebra from Hungerford's GTM73 (Ch 1-4); Field Theory from GTM167 (almost all the book); CA from Atiyah (Ch 1-9); A bit representation theory from GTM42, Part 1; No knowledge of HA and number theory.
Topology and geometry: Some knowledge of AT, mainly fundamental groups and covering spaces; A bit differential manifolds from GTM218 Ch 1-4, half of Ch 5.